Problem: All of the 4th grade teachers and students from Oak went on a field trip to an art museum. Tickets were $$5.00$ each for teachers and $$2.50$ each for students, and the group paid $$25.00$ in total. The next month, the same group visited a science museum where the tickets cost $$20.00$ each for teachers and $$8.50$ each for students, and the group paid $$91.00$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5x+2.5y = 25}$ ${20x+8.5y = 91}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-20x-10y = -100}$ ${20x+8.5y = 91}$ Add the top and bottom equations together. $ -1.5y = -9 $ $ y = \dfrac{-9}{-1.5}$ ${y = 6}$ Now that you know ${y = 6}$ , plug it back into $ {5x+2.5y = 25}$ to find $x$ ${5x + 2.5}{(6)}{= 25}$ $5x+15 = 25$ $5x = 10$ $x = \dfrac{10}{5}$ ${x = 2}$ You can also plug ${y = 6}$ into $ {20x+8.5y = 91}$ and get the same answer for $x$ ${20x + 8.5}{(6)}{= 91}$ ${x = 2}$ There were $2$ teachers and $6$ students on the field trips.